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Theorem dchrisum0fmul 20651
Description: The function  F, the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum2.g  |-  G  =  (DChr `  N )
rpvmasum2.d  |-  D  =  ( Base `  G
)
rpvmasum2.1  |-  .1.  =  ( 0g `  G )
dchrisum0f.f  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
dchrisum0f.x  |-  ( ph  ->  X  e.  D )
dchrisum0fmul.a  |-  ( ph  ->  A  e.  NN )
dchrisum0fmul.b  |-  ( ph  ->  B  e.  NN )
dchrisum0fmul.m  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
Assertion
Ref Expression
dchrisum0fmul  |-  ( ph  ->  ( F `  ( A  x.  B )
)  =  ( ( F `  A )  x.  ( F `  B ) ) )
Distinct variable groups:    q, b,
v, A    N, q    B, b, q, v    L, b, v    X, b, v
Allowed substitution hints:    ph( v, q, b)    D( v, q, b)    .1. ( v, q, b)    F( v, q, b)    G( v, q, b)    L( q)    N( v, b)    X( q)    Z( v, q, b)

Proof of Theorem dchrisum0fmul
Dummy variables  k 
i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrisum0fmul.a . . 3  |-  ( ph  ->  A  e.  NN )
2 dchrisum0fmul.b . . 3  |-  ( ph  ->  B  e.  NN )
3 dchrisum0fmul.m . . 3  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
4 eqid 2284 . . 3  |-  { q  e.  NN  |  q 
||  A }  =  { q  e.  NN  |  q  ||  A }
5 eqid 2284 . . 3  |-  { q  e.  NN  |  q 
||  B }  =  { q  e.  NN  |  q  ||  B }
6 eqid 2284 . . 3  |-  { q  e.  NN  |  q 
||  ( A  x.  B ) }  =  { q  e.  NN  |  q  ||  ( A  x.  B ) }
7 rpvmasum2.g . . . 4  |-  G  =  (DChr `  N )
8 rpvmasum.z . . . 4  |-  Z  =  (ℤ/n `  N )
9 rpvmasum2.d . . . 4  |-  D  =  ( Base `  G
)
10 rpvmasum.l . . . 4  |-  L  =  ( ZRHom `  Z
)
11 dchrisum0f.x . . . . 5  |-  ( ph  ->  X  e.  D )
1211adantr 451 . . . 4  |-  ( (
ph  /\  j  e.  { q  e.  NN  | 
q  ||  A }
)  ->  X  e.  D )
13 ssrab2 3259 . . . . . . 7  |-  { q  e.  NN  |  q 
||  A }  C_  NN
1413sseli 3177 . . . . . 6  |-  ( j  e.  { q  e.  NN  |  q  ||  A }  ->  j  e.  NN )
1514nnzd 10112 . . . . 5  |-  ( j  e.  { q  e.  NN  |  q  ||  A }  ->  j  e.  ZZ )
1615adantl 452 . . . 4  |-  ( (
ph  /\  j  e.  { q  e.  NN  | 
q  ||  A }
)  ->  j  e.  ZZ )
177, 8, 9, 10, 12, 16dchrzrhcl 20480 . . 3  |-  ( (
ph  /\  j  e.  { q  e.  NN  | 
q  ||  A }
)  ->  ( X `  ( L `  j
) )  e.  CC )
1811adantr 451 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  B }
)  ->  X  e.  D )
19 ssrab2 3259 . . . . . . 7  |-  { q  e.  NN  |  q 
||  B }  C_  NN
2019sseli 3177 . . . . . 6  |-  ( k  e.  { q  e.  NN  |  q  ||  B }  ->  k  e.  NN )
2120nnzd 10112 . . . . 5  |-  ( k  e.  { q  e.  NN  |  q  ||  B }  ->  k  e.  ZZ )
2221adantl 452 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  B }
)  ->  k  e.  ZZ )
237, 8, 9, 10, 18, 22dchrzrhcl 20480 . . 3  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  B }
)  ->  ( X `  ( L `  k
) )  e.  CC )
2415, 21anim12i 549 . . . 4  |-  ( ( j  e.  { q  e.  NN  |  q 
||  A }  /\  k  e.  { q  e.  NN  |  q  ||  B } )  ->  (
j  e.  ZZ  /\  k  e.  ZZ )
)
2511adantr 451 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ZZ  /\  k  e.  ZZ ) )  ->  X  e.  D )
26 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ZZ  /\  k  e.  ZZ ) )  -> 
j  e.  ZZ )
27 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ZZ  /\  k  e.  ZZ ) )  -> 
k  e.  ZZ )
287, 8, 9, 10, 25, 26, 27dchrzrhmul 20481 . . . . 5  |-  ( (
ph  /\  ( j  e.  ZZ  /\  k  e.  ZZ ) )  -> 
( X `  ( L `  ( j  x.  k ) ) )  =  ( ( X `
 ( L `  j ) )  x.  ( X `  ( L `  k )
) ) )
2928eqcomd 2289 . . . 4  |-  ( (
ph  /\  ( j  e.  ZZ  /\  k  e.  ZZ ) )  -> 
( ( X `  ( L `  j ) )  x.  ( X `
 ( L `  k ) ) )  =  ( X `  ( L `  ( j  x.  k ) ) ) )
3024, 29sylan2 460 . . 3  |-  ( (
ph  /\  ( j  e.  { q  e.  NN  |  q  ||  A }  /\  k  e.  { q  e.  NN  |  q 
||  B } ) )  ->  ( ( X `  ( L `  j ) )  x.  ( X `  ( L `  k )
) )  =  ( X `  ( L `
 ( j  x.  k ) ) ) )
31 fveq2 5486 . . . 4  |-  ( i  =  ( j  x.  k )  ->  ( L `  i )  =  ( L `  ( j  x.  k
) ) )
3231fveq2d 5490 . . 3  |-  ( i  =  ( j  x.  k )  ->  ( X `  ( L `  i ) )  =  ( X `  ( L `  ( j  x.  k ) ) ) )
331, 2, 3, 4, 5, 6, 17, 23, 30, 32fsumdvdsmul 20431 . 2  |-  ( ph  ->  ( sum_ j  e.  {
q  e.  NN  | 
q  ||  A } 
( X `  ( L `  j )
)  x.  sum_ k  e.  { q  e.  NN  |  q  ||  B } 
( X `  ( L `  k )
) )  =  sum_ i  e.  { q  e.  NN  |  q  ||  ( A  x.  B
) }  ( X `
 ( L `  i ) ) )
34 rpvmasum.a . . . . 5  |-  ( ph  ->  N  e.  NN )
35 rpvmasum2.1 . . . . 5  |-  .1.  =  ( 0g `  G )
36 dchrisum0f.f . . . . 5  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
378, 10, 34, 7, 9, 35, 36dchrisum0fval 20650 . . . 4  |-  ( A  e.  NN  ->  ( F `  A )  =  sum_ j  e.  {
q  e.  NN  | 
q  ||  A } 
( X `  ( L `  j )
) )
381, 37syl 15 . . 3  |-  ( ph  ->  ( F `  A
)  =  sum_ j  e.  { q  e.  NN  |  q  ||  A } 
( X `  ( L `  j )
) )
398, 10, 34, 7, 9, 35, 36dchrisum0fval 20650 . . . 4  |-  ( B  e.  NN  ->  ( F `  B )  =  sum_ k  e.  {
q  e.  NN  | 
q  ||  B } 
( X `  ( L `  k )
) )
402, 39syl 15 . . 3  |-  ( ph  ->  ( F `  B
)  =  sum_ k  e.  { q  e.  NN  |  q  ||  B } 
( X `  ( L `  k )
) )
4138, 40oveq12d 5838 . 2  |-  ( ph  ->  ( ( F `  A )  x.  ( F `  B )
)  =  ( sum_ j  e.  { q  e.  NN  |  q  ||  A }  ( X `  ( L `  j
) )  x.  sum_ k  e.  { q  e.  NN  |  q  ||  B }  ( X `  ( L `  k
) ) ) )
421, 2nnmulcld 9789 . . 3  |-  ( ph  ->  ( A  x.  B
)  e.  NN )
438, 10, 34, 7, 9, 35, 36dchrisum0fval 20650 . . 3  |-  ( ( A  x.  B )  e.  NN  ->  ( F `  ( A  x.  B ) )  = 
sum_ i  e.  {
q  e.  NN  | 
q  ||  ( A  x.  B ) }  ( X `  ( L `  i ) ) )
4442, 43syl 15 . 2  |-  ( ph  ->  ( F `  ( A  x.  B )
)  =  sum_ i  e.  { q  e.  NN  |  q  ||  ( A  x.  B ) }  ( X `  ( L `  i )
) )
4533, 41, 443eqtr4rd 2327 1  |-  ( ph  ->  ( F `  ( A  x.  B )
)  =  ( ( F `  A )  x.  ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   {crab 2548   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5820   1c1 8734    x. cmul 8738   NNcn 9742   ZZcz 10020   sum_csu 12154    || cdivides 12527    gcd cgcd 12681   Basecbs 13144   0gc0g 13396   ZRHomczrh 16447  ℤ/nczn 16450  DChrcdchr 20467
This theorem is referenced by:  dchrisum0flblem2  20654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-ec 6658  df-qs 6662  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-rp 10351  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-dvds 12528  df-gcd 12682  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-0g 13400  df-imas 13407  df-divs 13408  df-mnd 14363  df-mhm 14411  df-grp 14485  df-minusg 14486  df-sbg 14487  df-mulg 14488  df-subg 14614  df-nsg 14615  df-eqg 14616  df-ghm 14677  df-cmn 15087  df-abl 15088  df-mgp 15322  df-rng 15336  df-cring 15337  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-rnghom 15492  df-subrg 15539  df-lmod 15625  df-lss 15686  df-lsp 15725  df-sra 15921  df-rgmod 15922  df-lidl 15923  df-rsp 15924  df-2idl 15980  df-cnfld 16374  df-zrh 16451  df-zn 16454  df-dchr 20468
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